Analysis I show that lim_(n→∞) (1+(z/n))^n =e^z . uniformly convergence for all 𝛜>0 there exist N∈N whenever z∈E such that N<n ⇒ ∣f_n (z)−f(z)∣<𝛜 ∼Weiestrass M-test∼ Suppose that {f_n ^ }_(n=1) ^∞ is a sequence real valued function defined on a set E,and that there is a sequence of non-negetive number M_n satisfying the conditions for all n∈N , ∣f_n ^ (x)∣<M_n on a set E more specifically, ∣Σ_(h=1) ^∞ f_h (x)∣<Σ_(h=1) ^∞ ∣f_h (x)∣<Σ_(h=1) ^∞ M_h (Σ_(h=1) ^∞ M_h is convergent to an any arbitrary const.) then, Σ_(h=1) ^∞ f_h (x) is uniformly convergence. Analysis II prove f(t)= (2/π)−(4/π)∙Σ_(k=1) ^∞ ((cos(2kt))/(4k^2 −1)) uniformly convervence where t∈[−π,π] Analysis III Show that g(x)=Π^∞ _(n=1) (1−((x/(nπ)))^2 ) uniformly convergence. where x∈[−M,M]